Draw this isosceles triangle with its base being the side = 12.
Then, the left side is 10, and the right side is also 10.
Drop a line from the top vertex down to the middle of the base; it will be perpendicular to the base.
The left-hand triangle is a right triangle with a base of 6 and a hypotenuse of 12.
Using the Pythagorean Theorem: a2 + b2 = c2:
a = 6 c = 10 ---> 62 + b2 = 102 ---> 36 + b2 = 100 ---> b2 = 64
---> b = 8, which is the altitude to the base = 12
The area of a triangle can be found using the formula: A = ½ · b · h
The base of the triangle = 12 and the height = 8 ---> A = ½ · 12 · 8 = 48
Now, the area of the triangle on the left side of the triangle is one-half the total area = 24.
Using the formula for the area of the triangle for the triangle on the left side:
the base = 10 and the area = 24 ---> A = ½ · b · h ---> 48 = ½ · 10 · h
---> h = 9.6, which is the altitude to the left side.
Since the right-hand side triangle is congruent, the altitude to the right side is also 9.6.
Summing these three values: 12 + 9.6 + 9.6 = 31.2
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